All right, I'm back home and took a look at my Kazakh textbook--it only has the "correct" (non-Russified) structure for possession ("Mening mysyghym bar = I have a cat).
The other form I was thinking of was that for expressing need--with either nouns or verbs. With nouns, the needer is in the dative, and the thing needed in the nominative, followed by the word kerek, indicating need, or kerek yemes, indicating lack of need:
Maghan nan kerek (yemes).
I-DAT bread-NOM need (not)
"I need (don't need) bread." (literally, "To me bread is (not) needed."
...which is the same structure as in Russian (Mne kleb nyzhen/ne nyzhen).
With verbs, on the other hand, the possessive form of the verbal noun (which also serves as the infinitive) is used with kerek, and the verb itself is negated:
Mening baruym (barmauym) kerek.
I-GEN [to go]-1p/POSS ([to go]-NEG-1p/POSS) need
"I need (don't need) to go"; literally, "My going (not going) is needed."
However, I have definitely also encountered the dative structure used with the verb as well:
Maghan bary kerek (yemes).
I-DAT [to go] need (not)
"I need (don't need) to go" = "To me going is (not) needed."
...which matches the Russian (Mne nado idti).
[Somewhere I was also taught about attaching personal endings to kerek: keregim "my need," kereging "your needed"--but I don't recall where they were used.]
The reason I make a big deal of this is that it's a clear case of one language lifting a piece of grammar from another, which is something I didn't think much about previously or know how to handle. I knew that Kazakh had borrowed many words from Russian and created Kazakh calques of Russian words (temir zohly, or "iron road" for railway, from the Russian zheleznaya doroga, which means the same thing), but I hadn't realized that there were grammatical borrowings as well (although I did notice the similarities).
This is something I'll have to keep in mind as I have my conlangs influence each other (although it might be difficult, seeing as how thus far my two most important language families are different in almost every way: VSO vs. SOV, prepositions vs. postpositions, head-first vs. head-final, gender vs. no gender, etc.). Any idea how two languages like that might mix?